Topology optimization (TO) has experienced a dramatic development over the last decades aided by the arising of metamaterials and additive manufacturing (AM) techniques, and it is intended to achieve the current and future challenges. In this paper we propose an extension for linear orthotropic materials of a three-dimensional TO algorithm which directly operates on the six elastic properties -- three longitudinal and shear moduli, having fixed three Poisson ratios -- of the finite element (FE) discretization of certain analysis domain. By performing a gradient-descent-alike optimization on these properties, the standard deviation of a strain-energy measurement is minimized, thus coming up with optimized, strain-homogenized structures with variable longitudinal and shear stiffness in their different material directions. To this end, an orthotropic formulation with two approaches -- direct or strain-based and complementary or stress-based -- has been developed for this optimization problem, being the stress-based more efficient as previous works on this topic have shown. The key advantages that we propose are: (1) the use of orthotropic ahead of isotropic materials, which enables a more versatile optimization process since the design space is increased by six times, and (2) no constraint needs to be imposed (such as maximum volume) in contrast to other methods widely used in this field such as Solid Isotropic Material with Penalization (SIMP), all of this by setting one unique hyper-parameter. Results of four designed load cases show that this orthotropic-TO algorithm outperforms the isotropic case, both for the similar algorithm from which this is an extension and for a SIMP run in a FE commercial software, presenting a comparable computational cost. We remark that it works particularly effectively on pure shear or shear-governed problems such as torsion loading.

翻译：拓扑优化（TO）在过去几十年间因超材料和增材制造（AM）技术的兴起而经历了迅猛发展，旨在应对当前及未来的挑战。本文提出了一种针对线性正交各向异性材料的三维拓扑优化算法的扩展方案，该算法直接作用于特定分析域有限元（FE）离散单元的六项弹性属性——即三个纵向模量与三个剪切模量（泊松比固定为三个）。通过对这些属性执行类似梯度下降的优化，应变能测量的标准差被最小化，从而在材料的不同方向上生成具有可变纵向刚度与剪切刚度的优化均质化应变结构。为此，本文针对该优化问题开发了两种正交各向异性公式——直接法（基于应变）与互补法（基于应力），其中基于应力的方法效率更高（正如该领域先前研究所表明）。我们提出的关键优势在于：（1）采用正交各向异性替代各向同性材料，使设计空间扩大六倍，从而显著提升优化过程的灵活性；（2）无需像固体各向同性材料惩罚法（SIMP）等该领域广泛使用的其他方法那样施加约束（如最大体积），且仅需设定一个超参数。四个设计载荷工况的结果表明，此正交各向异性拓扑优化算法在性能上优于各向同性情形——无论是作为其扩展基础的相似算法，还是在有限元商业软件中运行的SIMP方法，且计算成本相当。值得注意的是，该算法在纯剪切或剪切主导问题（如扭转载荷）中尤为有效。